U.W. Bangor - School of Informatics - Mathematics Preprints 2002

Algebraic Topology

The intuitions of higher dimensional algebra for the study of structured space

Summary:

Higher dimensional algebra frees mathematics from the restriction to a
purely linear notation, in order to improve the modelling of geometry
and so obtain more understanding and more modes of computation.
It gives new tools for non-commutative, higher dimensional,
local to global problems, through the notion of
algebraic inverse to subdivision.
We explain the way these ideas arose for the writers,
in extending first the classical notion of abstract group to abstract groupoid,
in which composition is only partially defined, as in composing journeys,
and which brings a spatial component to the usual group theory.
An example from knot theory is is used to explain how such algebra
can be used to describe some structure of a space.
The extension to dimension 2 uses compositions of squares
in two directions, and the richness of the resulting algebra
is shown by some 2-dimensional calculations.
The difficulty of the jump from dimension 1 to dimension 2 is also
illustrated by the comparison of the commutative square
with the commutative cube - discussion of the latter requires new ideas.
The importance of category theory is explained,
and a range of current and potential applications
of higher dimensional algebra is indicated.

Crossed squares and 2-crossed modules

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Crossed complexes, and free crossed resolutions
for amalgamated sums and HNN-extensions of groups

Summary:

The category of crossed complexes gives an algebraic model of
CW-complexes and cellular maps. Free crossed resolutions of
groups contain information on a presentation of the group as well
as higher homological information. We relate this to the problem
of calculating non-abelian extensions. We show how the strong
properties of this category allow for the computation of free
crossed resolutions for amalgamated sums and HNN-extensions of
groups, and so obtain computations of higher homotopical syzygies
in these cases.

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Connections, local subgroupoids, and a holonomy
Lie groupoid of a line bundle gerbe

Abstract:

Our main aim is to associate a holonomy Lie groupoid to the
connective structure of an abelian gerbe.
The construction has analogies with a procedure for the holonomy Lie groupoid
of a foliation, in using a locally Lie groupoid and a globalisation procedure.
We show that path connections and 2-holonomy on line
bundles may be formulated using the notion of a connection pair on
a double category, due to Brown-Spencer, but now formulated in
terms of double groupoids using the thin fundamental groupoids
introduced by Caetano-Mackaay-Picken.
To obtain a locally Lie groupoid to which globalisation applies,
we use methods of local subgroupoids as developed by Brown-Icen-Mucuk.

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Cubical abelian groups with connections are equivalent to chain complexes

Abstract:

The theorem of the title is shown to be a consequence of the
equivalence between crossed complexes and cubical
omega-groupoids with connections proved by us in [BH3].
We assume the definitions given in [BH3]. Thus this paper is a
companion to others, for example [T1], which show that a
deficit of the traditional theory of cubical sets and cubical
groups has been the lack of attention paid to the ``connections'',
defined in [BH3]. Indeed the traditional degeneracies of
cubical theory identify certain opposite faces of a cube, unlike
the degeneracies of simplicial theory which identify adjacent
faces. The connections allow for a fuller analogy with the methods
available for simplicial theory by giving forms of `degeneracies'
which identify adjacent faces of cubes. They are used in
[BH3] and [ABS] to give a definition of a `commutative cube'.

Part of the interest of these results is that the family of
categories equivalent to that of crossed complexes can be regarded
as a foundation for a non-abelian approach to algebraic topology
and the cohomology of groups. These results show that a form of
abelianisation of these categories leads to well-known structures.

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The fundamental groupoid of the quotient of a Hausdorff space by a
discontinuous action of a discrete group is the orbit groupoid

Abstract:

The main result is that the fundamental groupoid of the orbit
space of a discontinuous action of a discrete group on a
Hausdorff space is the orbit groupoid of the fundamental groupoid
of the space. This result, which is related to work of Armstrong,
is due to Brown and Higgins in 1985 and was published in sections
9 and 10 of Chapter 9 of the first author's book on Topology
[Brown:1988]. Since the book is out of print, and the result
seems not well known, we now advertise it here. We also describe
work of Higgins and of Taylor which makes this result usable for
calculations. As an example, we compute the fundamental group of
the symmetric square of a space.

This is a somewhat edited, and in one point (on normal closures)
corrected, version of those sections of [Brown:1988]. It is
also hoped that this publication will allow wider views of this
result, for example in topos theory and descent theory.

Because of its provenance, this should be read as a graduate text
rather than an article. This explains also the inclusion of
exercises. It is expected that this material will be part of a new
edition of the book.

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Abstract:

(This is an extended account of a lecture given at the meeting on
`Categorical Structures for Descent and Galois Theory,
Hopf Algebras, and Semiabelian Categories',
Fields Institute, September 23-28, 2002.)

We outline the main features of the definitions and applications of
crossed complexes and cubical omega-groupoids with connections.
These give forms of higher homotopy groupoids,
and new views of basic algebraic topology and the cohomology of groups,
with the ability to obtain non commutative results
and compute homotopy types.